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Mathematical definitions

Irradiance

Irradiance of a surface, denoted Ee ("e" for "energetic", to avoid confusion with photometric quantities), is defined as²

E e = ∂ Φ e ∂ A , {\displaystyle E_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}},}

where

• ∂ is the partial derivative symbol;
• Φe is the radiant flux received;
• A is the area.

The radiant flux emitted by a surface is called radiant exitance.

Spectral irradiance

Spectral irradiance in frequency of a surface, denoted Ee,ν, is defined as³

E e , ν = ∂ E e ∂ ν , {\displaystyle E_{\mathrm {e} ,\nu }={\frac {\partial E_{\mathrm {e} }}{\partial \nu }},}

where ν is the frequency.

Spectral irradiance in wavelength of a surface, denoted Ee,λ, is defined as⁴

E e , λ = ∂ E e ∂ λ , {\displaystyle E_{\mathrm {e} ,\lambda }={\frac {\partial E_{\mathrm {e} }}{\partial \lambda }},}

where λ is the wavelength.

Property

Irradiance of a surface is also, according to the definition of radiant flux, equal to the time-average of the component of the Poynting vector perpendicular to the surface:

E e = ⟨ | S | ⟩ cos ⁡ α , {\displaystyle E_{\mathrm {e} }=\langle |\mathbf {S} |\rangle \cos \alpha ,}

where

• ⟨ • ⟩ is the time-average;
• S is the Poynting vector;
• α is the angle between a unit vector normal to the surface and S.

For a propagating sinusoidal linearly polarized electromagnetic plane wave, the Poynting vector always points to the direction of propagation while oscillating in magnitude. The irradiance of a surface is then given by⁵

E e = n 2 μ 0 c E m 2 cos ⁡ α = n ε 0 c 2 E m 2 cos ⁡ α o r n 2 Z 0 E m 2 cos ⁡ α , {\displaystyle E_{\mathrm {e} }={\frac {n}{2\mu _{0}\mathrm {c} }}E_{\mathrm {m} }^{2}\cos \alpha ={\frac {n\varepsilon _{0}\mathrm {c} }{2}}E_{\mathrm {m} }^{2}\cos \alpha \,or{\frac {n}{2Z_{0}}}E_{\mathrm {m} }^{2}\cos \alpha ,}

where

• Em is the amplitude of the wave's electric field;
• n is the refractive index of the medium of propagation;
• c is the speed of light in vacuum;

• μ0 is the vacuum permeability;

• ε0 is the vacuum permittivity;
  • c = 1 ε 0 μ 0 {\textstyle c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}}
  • Z 0 = μ 0 c {\textstyle Z_{0}=\mu _{0}c} is the impedance of free space.

This formula assumes that the magnetic susceptibility is negligible; i.e. that μr ≈ 1 (μ ≈ μ0) where μr is the relative magnetic permeability of the propagation medium. This assumption is typically valid in transparent media in the optical frequency range.

Point source

A point source of light produces spherical wavefronts. The irradiance in this case varies inversely with the square of the distance from the source.

E = P A = P 4 π r 2 , {\displaystyle E={\frac {P}{A}}={\frac {P}{4\pi r^{2}}},}

where

• r is the distance;
• P is the radiant flux;
• A is the surface area of a sphere of radius r.

For quick approximations, this equation indicates that doubling the distance reduces irradiation to one quarter; or similarly, to double irradiation, reduce the distance to 71%.

In astronomy, stars are routinely treated as point sources even though they are much larger than the Earth. This is a good approximation because the distance from even a nearby star to the Earth is much larger than the star's diameter. For instance, the irradiance of Alpha Centauri A (radiant flux: 1.5 L☉, distance: 4.34 ly) is about 2.7 × 10−8 W/m2 on Earth.

Solar irradiance

Main article: Solar irradiance

The global irradiance on a horizontal surface on Earth consists of the direct irradiance Ee,dir and diffuse irradiance Ee,diff. On a tilted plane, there is another irradiance component, Ee,refl, which is the component that is reflected from the ground. The average ground reflection is about 20% of the global irradiance. Hence, the irradiance Ee on a tilted plane consists of three components:⁶

E e = E e , d i r + E e , d i f f + E e , r e f l . {\displaystyle E_{\mathrm {e} }=E_{\mathrm {e} ,\mathrm {dir} }+E_{\mathrm {e} ,\mathrm {diff} }+E_{\mathrm {e} ,\mathrm {refl} }.}

The integral of solar irradiance over a time period is called "solar exposure" or "insolation".⁷⁸

Average solar irradiance at the top of the Earth's atmosphere is roughly 1361 W/m2, but at surface irradiance is approximately 1000 W/m2 on a clear day.

SI radiometry units

SI radiometry units

• v
• t
• e

Quantity		Unit		Dimension	Notes
Name	Symbol9	Name	Symbol
Radiant energy	Qe10	joule	J	M⋅L2⋅T−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m3	M⋅L−1⋅T−2	Radiant energy per unit volume.
Radiant flux	Φe11	watt	W = J/s	M⋅L2⋅T−3	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and called luminosity in astronomy.
Spectral flux	Φe,ν12	watt per hertz	W/Hz	M⋅L2⋅T −2	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.
	Φe,λ13	watt per metre	W/m	M⋅L⋅T−3
Radiant intensity	Ie,Ω14	watt per steradian	W/sr	M⋅L2⋅T−3	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity	Ie,Ω,ν15	watt per steradian per hertz	W⋅sr−1⋅Hz−1	M⋅L2⋅T−2	Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.
	Ie,Ω,λ16	watt per steradian per metre	W⋅sr−1⋅m−1	M⋅L⋅T−3
Radiance	Le,Ω17	watt per steradian per square metre	W⋅sr−1⋅m−2	M⋅T−3	Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
Spectral radianceSpecific intensity	Le,Ω,ν18	watt per steradian per square metre per hertz	W⋅sr−1⋅m−2⋅Hz−1	M⋅T−2	Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
	Le,Ω,λ19	watt per steradian per square metre, per metre	W⋅sr−1⋅m−3	M⋅L−1⋅T−3
IrradianceFlux density	Ee20	watt per square metre	W/m2	M⋅T−3	Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradianceSpectral flux density	Ee,ν21	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10−26 W⋅m−2⋅Hz−1) and solar flux unit (1 sfu = 10−22 W⋅m−2⋅Hz−1 = 104 Jy).
	Ee,λ22	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiosity	Je23	watt per square metre	W/m2	M⋅T−3	Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity	Je,ν24	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity".
	Je,λ25	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exitance	Me26	watt per square metre	W/m2	M⋅T−3	Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity".
Spectral exitance	Me,ν27	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity".
	Me,λ28	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exposure	He	joule per square metre	J/m2	M⋅T−2	Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure	He,ν29	joule per square metre per hertz	J⋅m−2⋅Hz−1	M⋅T−1	Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
	He,λ30	joule per square metre, per metre	J/m3	M⋅L−1⋅T−2
See also: SI Radiometry Photometry

See also

• Albedo
• Fluence
• Illuminance
• Insolation
• Light diffusion
• PI curve (photosynthesis-irradiance curve)
• Solar azimuth angle
• Solar irradiance
• Solar noon
• Spectral flux density
• Stefan–Boltzmann law

References

1. Carroll, Bradley W. (2017-09-07). An introduction to modern astrophysics. Cambridge University Press. p. 60. ISBN 978-1-108-42216-1. OCLC 991641816. 978-1-108-42216-1 ↩

2. "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15. http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943 ↩

3. "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15. http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943 ↩

4. "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15. http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943 ↩

5. Griffiths, David J. (1999). Introduction to electrodynamics (3. ed., reprint. with corr. ed.). Upper Saddle River, NJ [u.a.]: Prentice-Hall. ISBN 0-13-805326-X. 0-13-805326-X ↩

6. Quaschning, Volker (2003). "Technology fundamentals—The sun as an energy resource". Renewable Energy World. 6 (5): 90–93. /wiki/Volker_Quaschning ↩

7. Quaschning, Volker (2003). "Technology fundamentals—The sun as an energy resource". Renewable Energy World. 6 (5): 90–93. /wiki/Volker_Quaschning ↩

8. Liu, B. Y. H.; Jordan, R. C. (1960). "The interrelationship and characteristic distribution of direct, diffuse and total solar radiation". Solar Energy. 4 (3): 1. Bibcode:1960SoEn....4....1L. doi:10.1016/0038-092X(60)90062-1. /wiki/Bibcode_(identifier) ↩

9. Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities. /wiki/Standards_organization ↩

10. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance. ↩

11. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance. ↩

12. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩

13. Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩

14. Directional quantities are denoted with suffix "Ω". /wiki/%CE%A9 ↩

15. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩

16. Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩

17. Directional quantities are denoted with suffix "Ω". /wiki/%CE%A9 ↩

18. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩

19. Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩

20. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance. ↩

21. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩

22. Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩

23. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance. ↩

24. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩

25. Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩

26. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance. ↩

27. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩

28. Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩

29. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩

30. Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩